a right-atngled is such that the two shorter sides are (2x+1)centimetre and (5x-3) centimetre.If the area of the triangle is 42 centimetre^2,find the lengths of the two sides
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a right-angled is such that the two shorter sides are (2x+1)centimetre and (5x-3) centimetre.If the area of the triangle is 42 centimetre^2,find the lengths of the two sides
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Assuming you are talking about a right-angled triangle ...
area = (1/2) base x height
42 = (1/2)(2x+1)(5x-3)
84 = 10x^2 - x - 3
10x^2 - x - 87 = 0
(x - 3)(10x + 29) = 0
x = 3 or x = -19, but x = -29 would give us negative sides, so ...
x = 3 gives us sides of 7 and 12
check: (1/2)(7)(12) = 42 , looks good
To find the lengths of the two sides of the right-angled triangle, we can use the Pythagorean Theorem. The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Let's use this theorem to solve the problem. We have the two shorter sides of the triangle, which are (2x+1) centimeters and (5x-3) centimeters. Let's assume these are the two legs of the triangle.
Using the Pythagorean Theorem, we have:
(2x+1)^2 + (5x-3)^2 = hypotenuse^2
Expanding and simplifying:
4x^2 + 4x + 1 + 25x^2 - 30x + 9 = hypotenuse^2
Combining like terms:
29x^2 - 26x + 10 = hypotenuse^2
We know that the area of the triangle is 42 square centimeters, so we can set up another equation:
(1/2) * (2x + 1) * (5x - 3) = 42
Expanding and simplifying:
(5x^2 - x - 3) = 42
Rearranging:
5x^2 - x - 45 = 0
We can solve this quadratic equation to find the value of x. Once we find the value of x, we can substitute it back into the expressions for the two shorter sides (2x+1) and (5x-3) to find their lengths.
To find the lengths of the two sides of a right-angled triangle, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the two shorter sides are (2x + 1) centimeters and (5x - 3) centimeters. Let's assume that the hypotenuse is of length h centimeters.
According to the Pythagorean theorem, we have:
(2x + 1)^2 + (5x - 3)^2 = h^2
Expanding and simplifying the equation, we get:
4x^2 + 4x + 1 + 25x^2 - 30x + 9 = h^2
Combining like terms, we have:
29x^2 - 26x + 10 = h^2
Now, we are given that the area of the triangle is 42 square centimeters. The formula for calculating the area of a right-angled triangle is (1/2) * base * height.
In this case, the base and height are (2x + 1) centimeters and (5x - 3) centimeters, respectively. So we have:
(1/2) * (2x + 1) * (5x - 3) = 42
Simplifying this equation, we have:
(2x + 1)(5x - 3) = 84
Expanding the equation, we get:
10x^2 - 3x - 17 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula.
Factoring, we have:
(5x - 7)(2x + 2) = 0
Setting each factor equal to zero, we get:
5x - 7 = 0 or 2x + 2 = 0
Solving these equations, we find:
x = 7/5 or x = -1
Since length cannot be negative, we discard x = -1.
Now, we can substitute x = 7/5 into our equations to find the lengths of the two sides.
The first side is:
2x + 1 = 2(7/5) + 1 = 14/5 + 5/5 = 19/5 centimeters
The second side is:
5x - 3 = 5(7/5) - 3 = 7 - 3 = 4 centimeters
Therefore, the lengths of the two sides of the right-angled triangle are 19/5 centimeters and 4 centimeters.